Taiwan's President Tsai Ing-wen, who was inaugurated for a second term yesterday, said the island cannot accept becoming part of China under "one country, two systems".
This is certain to set the stage for ever worsening cross-strait ties.
Washington's congratulatory message to Ms Tsai has also triggered a warning from Beijing, which vowed to "take necessary countermeasures" in response, amid mounting tensions between the United States and China over the handling of the coronavirus pandemic.
US Secretary of State Mike Pompeo, referring to Ms Tsai as "president" in a congratulatory note, hailed her "courage and wisdom".
"Support for Taiwan in the United States is bipartisan and unanimous, as demonstrated by the recent enactment of the Taipei Act which strengthens our overall relationship and supports a closer economic partnership," he said.
Mr Pompeo was the first sitting US secretary of state to congratulate an incoming Taiwanese leader, sparking a rebuke from Beijing, which bristles at any formal recognition of Taipei.
Taiwan's Presidential Office expressed its thanks to Mr Pompeo in a statement.
"The message testifies to strong US support for Taiwan's democracy and underscores Taiwan-US mutual trust and goodwill," said Mr Alex Huang, the Presidential Office spokesman.
China's Cabinet-level Taiwan Affairs Office, Ministry of Defence and Ministry of Foreign Affairs all responded to Mr Pompeo's message via statements published on their official social media.
China's Defence Ministry called the Taiwan issue "China's internal affair" and said it opposes "any form of official exchanges and military contacts between Taiwan and any country".
It added: "Relying on foreigners to raise one's profile is a road that leads nowhere, and using the Taiwan issue to pin down China is doomed to fail."
Support for Taiwan in the United States is bipartisan and unanimous, as demonstrated by the recent enactment of the Taipei Act which strengthens our overall relationship and supports a closer economic partnership.
U.S. SECRETARY OF STATE MIKE POMPEO
Threatening the US with "countermeasures" in response to its "erroneous actions", the Ministry of Foreign Affairs said: "China urges the US side to immediately correct its mistakes."
The Ministry of National Defence said that the military would "take all necessary measures to firmly safeguard" China's sovereignty.
China sees Taiwan as a renegade province to be reunified with the mainland.
In a speech after being sworn in for her final term in office, Ms Tsai said Taiwan's relationship with China has reached a historical turning point, adding that "both sides have a duty to find a way to coexist over the long term and prevent the intensification of antagonism and differences".
Ms Tsai and Vice-President William Lai were sworn into office in a closed-door ceremony at the Presidential Office due to the Covid-19 outbreak.
Ms Tsai reiterated her stance, saying she "will not accept the Beijing authorities' use of 'one country, two systems' policy to downgrade Taiwan and undermine the cross-strait status quo".
Taiwan's main opposition party, the China-friendly Kuomintang, referring to Taiwan by its formal name, told The Straits Times yesterday: "Tsai Ing-wen is the Republic of China's President for the next four years, elected by the people of the Republic of China… The Chinese Communist Party should acknowledge the existence of the Republic of China to further promote cross-strait exchanges."
Ms Tsai also said that Taiwan will continue its fight to participate in international organisations, and "bolster ties with the US, Japan, Europe, and other like-minded countries".
The Trump administration had thrown its weight behind an international campaign to grant Taiwan access to the World Health Assembly earlier this week.
Mr Pompeo, at a State Department briefing later yesterday, again congratulated Ms Tsai on her inauguration. But he did not directly answer a question on US-Taiwan relations.
Mr Pompeo also hit back at Beijing, telling reporters that China was being run by a "brutal communist regime", adding that "China's contribution to fighting the pandemic is paltry compared to the cost it has imposed on the world".
- Additional reporting by Elizabeth Law